[Paper Review] Random conformal dynamical systems

Bertrand Deroin, Victor Kleptsyn|Archive ouverte UNIGE (University of Geneva)|Jun 10, 2005

Mathematical Dynamics and Fractals27 references4 citations

TL;DR

This paper establishes a dichotomy for random conformal dynamical systems: either there exists a probability measure invariant under all transformations in the pseudo-group, or almost surely, long random compositions contract exponentially. The authors prove unique ergodicity and equidistribution of orbits under the exponential contraction condition, extending results from symmetric to general random conformal systems using stochastic analysis and harmonic measures on foliations.

ABSTRACT

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversaly conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudo-group), or almost surely a long composition of maps contracts exponentially a ball. We deduce some results about the unique ergodicity.

Motivation & Objective

• To establish a dichotomy for random conformal dynamical systems acting on manifolds, generalizing known results from symmetric to non-symmetric settings.
• To investigate the existence of invariant measures under random compositions of conformal maps or holonomy pseudo-groups in transversely conformal foliations.
• To prove equidistribution of random orbits under the exponential contraction property, even in non-symmetric systems.
• To extend the theory of harmonic measures and Brownian motion on foliations to higher codimension and non-compact settings.
• To resolve the question of whether harmonic measures can have atoms by showing they must be diffuse under exponential contraction.

Proposed method

• Use of the Wiener measure and Brownian motion along leaves of a foliation to model random compositions of holonomy maps.
• Application of the heat kernel and diffusion semigroup $D^t$ to analyze the evolution of measures under random dynamics.
• Derivation of a key integral formula (5.21) relating conditional measures $\nu_x$ to transition densities of the Brownian motion via heat kernel quotients.
• Discretization of time to $t = k\delta$ and construction of a Markovian stopping time $\tau$ where holonomy maps exhibit exponential contraction.
• Estimation of the product of heat kernel ratios using bounds from Lemma 2.4 to show uniform lower bounds on measure transport.
• Use of superharmonicity and comparison with $D^t_*\mu$ to show that harmonic measures cannot have atoms, leading to contradiction if atoms exist.

Experimental results

Research questions

• RQ1 Under what conditions does a random conformal dynamical system admit a fully invariant probability measure?
• RQ2 What is the asymptotic behavior of random compositions of conformal maps when no invariant measure exists?
• RQ3 Can unique ergodicity be established for non-symmetric random conformal systems under exponential contraction?
• RQ4 How do harmonic measures behave under random holonomy in transversely conformal foliations, particularly regarding atomicity?
• RQ5 To what extent can the theory of Brownian motion and heat kernels be extended to higher codimension foliations with boundary?

Key findings

• A dichotomy holds: either there exists a probability measure invariant under all elements of the pseudo-group, or almost every random composition contracts a neighborhood exponentially.
• In the exponential contraction regime, the orbit of any point under random compositions is uniquely equidistributed with respect to a single invariant measure.
• The harmonic measure $\mu$ on the foliated space cannot have atoms, as shown by contradiction using the lower bound on heat kernel products.
• For sufficiently large $T$, the probability that a stopping time $\tau < 2T$ with exponential contraction is close to 1, ensuring the validity of the estimates.
• The estimate (5.23) shows that $\nu_x(I) \geq W_{x_0}(N) \cdot c_1 \cdot c_0 > 0$ for small $I$, contradicting atomicity of $\nu_x$.
• The results extend to higher codimension foliations by working in the compactified space $\overline{M}$, where superharmonicity and boundary behavior are controlled.